
theorem Th3:
  for V be ComplexAlgebra, V1 be ComplexSubAlgebra of V holds
  ( for v1,w1 be Element of V1, v,w be Element of V st
     v1=v & w1=w holds v1+w1=v+w ) &
  ( for v1,w1 be Element of V1, v,w be Element of V st
     v1=v & w1=w holds v1*w1=v*w ) &
  ( for v1 be Element of V1, v be Element of V, a be Complex st
     v1=v holds a*v1=a*v ) &
  1_V1 = 1_V & 0.V1=0.V
proof
  let V be ComplexAlgebra, V1 be ComplexSubAlgebra of V;
  hereby let x1,y1 be Element of V1, x,y be Element of V;
    assume
A1:x1=x & y1=y;
    x1 + y1 = ((the addF of V)||the carrier of V1).([x1,y1])
                         by Def1;
    hence x1 + y1 = x+y by A1,FUNCT_1:49;
  end;
  hereby let x1,y1 be Element of V1, x,y be Element of V;
    assume
A2:x1=x & y1=y;
    x1 * y1 = ((the multF of V)||the carrier of V1).([x1,y1])
                         by Def1;
    hence x1 * y1 = x*y by A2,FUNCT_1:49;
  end;
  hereby let v1 be Element of V1, v be Element of V, a be Complex;
    assume
A3:v1 = v;
    reconsider a1 = a as Element of COMPLEX by XCMPLX_0:def 2;
    a1 * v1 = ((the Mult of V) | [:COMPLEX,the carrier of V1:]).([a1,v1])
                               by Def1;
    then a1 * v1 = a1 * v by A3,FUNCT_1:49;
    hence a * v1 = a * v;
  end;
  thus thesis by Def1;
end;
